The Concepts of Natural Selection and Sexual Selection in Evolution

Darwin wasn't the first to come up with the idea that evolution took place; many others had thought that was the case, but had no reasonable explanation for how it could happen.
Darwin's amazing achievement was in coming up with the mechanism by which an unconscious nature could cause these changes to happen in ways that seemed, in hindsight, to be on a path toward a goal.
(Creationists still struggle to understand this despite Darwin's clear writing and painstakingly thorough scholarship.)
He came up with two methods of selection -- natural selection and sexual selection.
Some people think that sexual selection should be considered a subset of natural selection, but there are distinct differences between them.
The difference which I think is important is that the apparent (in hindsight) promotion of traits
through natural selection is actually just natural selection not killing them off (either directly
or through "genetic death, ie. the lineage dying out).
Natural selection doesn't cause a search for better or more optimal ways of doing things, it just lops
off those that really don't work well at a given time, even if those changes could be useful at some future time.
Natural selection, in essence, only works "against" the organism; everything "for" it is accidental, although the end result, in hindsight, looks like there was a direction.

Sexual selection, on the other hand, is generally selection "for" a feature or features, actively
chosen by another organism (though not necessarily consciously so).
It can also be active competition, usually between males, for instance either fighting or displaying, although there is often an element of choice by the non-fighting sex (usually females).

One of the strongest hints that a given feature is likely to be due to sexual selection is the timing of their appearance during the lifespan -- these traits typically aren't seen until the animal is able to mate.